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2003 Brownian Motion on Compact Manifolds: Cover Time and Late Points
Amir Dembo, Yuval Peres, Jay Rosen
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Electron. J. Probab. 8: 1-14 (2003). DOI: 10.1214/EJP.v8-139

Abstract

Let $M$ be a smooth, compact, connected Riemannian manifold of dimension $d \gt 2$ and without boundary. Denote by $T(x,r)$ the hitting time of the ball of radius $r$ centered at $x$ by Brownian motion on $M$. Then, $C_r(M)=\sup_{x \in M} T(x,r)$ is the time it takes Brownian motion to come within $r$ of all points in $M$. We prove that $C_r(M)/(r^{2-d}|\log r|)$ tends to $\gamma_d V(M)$ almost surely as $r\to 0$, where $V(M)$ is the Riemannian volume of $M$. We also obtain the ``multi-fractal spectrum'' $f(\alpha)$ for ``late points'', i.e., the dimension of the set of $\alpha$-late points $x$ in $M$ for which $\limsup_{r\to 0} T(x,r)/ (r^{2-d}|\log r|) = \alpha \gt 0$.

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Amir Dembo. Yuval Peres. Jay Rosen. "Brownian Motion on Compact Manifolds: Cover Time and Late Points." Electron. J. Probab. 8 1 - 14, 2003. https://doi.org/10.1214/EJP.v8-139

Information

Published: 2003
First available in Project Euclid: 23 May 2016

zbMATH: 1063.58021
MathSciNet: MR1998762
Digital Object Identifier: 10.1214/EJP.v8-139

Subjects:
Primary: 60J65

Keywords: Brownian motion , Cover time , Manifold , Wiener sausage

Rights: Copyright © 2003 The Institute of Mathematical Statistics and the Bernoulli Society

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